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Equivariant map
About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.
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TL;DR: The authors showed that the ideal for a general tree is generated by the ideals of its flattenings at vertices, not just an ideal defining the model set-theoretically.
Abstract: We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group based models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. The main novelty is our proof that this procedure yields the entire ideal, not just an ideal defining the model set-theoretically. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices.
54 citations
TL;DR: In this paper, the authors consider finite energy corotational wave maps with target manifold S2 and prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blowup case) or a linear scattering term up to an error which tends to 0 in the energy space.
Abstract: We consider finite energy corotational wave maps with target manifold S2. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blowup case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space. © 2015 Wiley Periodicals, Inc.
54 citations
01 Jan 2004
TL;DR: For a semi-simple simply connected algebraic group G, the authors showed that the corresponding Nekrasov partition function is basically equal to the Whittaker matrix coefficient in the universal Verma module over certain affine Lie algebra -the one whose root system is dual to that of the affinization of Lie(G).
Abstract: For a semi-simple simply connected algebraic group G we introduce certain parabolic analogues of the Nekrasov partition function (introduced by Nekrasov and studied recently by Nekrasov-Okounkov and Nakajima-Yoshioka for G=SL(n)). These functions count (roughly speaking) principal G-bundles on the projective plane with a trivialization at infinity and with a parabolic structure at the horizontal line. When the above parabolic subgroup is a Borel subgroup we show that the corresponding partition function is basically equal to the Whittaker matrix coefficient in the universal Verma module over certain affine Lie algebra - namely, the one whose root system is dual to that of the affinization of Lie(G). We explain how one can think about this result as the affine analogue of the results of Givental and Kim about Gromov-Witten invariants (more precisely, equivariant J-functions) of flag manifolds. Thus the main result of the paper may considered as the computation of the equivariant J-function of the affine flag manifold associated with G (in particular, we reprove the corresponding results for the usual flag manifolds) via the corresponding "Langlands dual" affine Lie algebra. As the main tool we use the algebro-geometric version of the Uhlenbeck space introduced by Finkelberg, Gaitsgory and the author. The connection of these results with the Seiberg-Witten prepotential will be treated in a subsequent publication.
54 citations
TL;DR: In this article, the authors studied the special case of toric Sasaki-Einstein manifolds known as Y p,q,p,q and derived the full perturbative part of the partition function in terms of a special function which appears to be a curious generalisation of the triple sine function.
Abstract: On any simply connected Sasaki–Einstein five dimensional manifold one can construct a super Yang–Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki–Einstein manifolds known as Y
p,q
manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of a certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large N behaviour for the case of single hypermultiplet in adjoint representation and we derive the N
3-behaviour in this case.
54 citations
TL;DR: In this paper, pure two-bubbles are constructed for energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale.
Abstract: We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k > 2. The concentrating speed of the second bubble is exponential for the first two models and a power function in the last case.
54 citations